JNL: PPCF PIPS: 337560 TYPE: SPE TS: NEWGEN DATE: 7/5/2010 EDITOR: MJ

IOP PUBLISHING PLASMA PHYSICS AND CONTROLLED FUSION

Plasma Phys. Control. Fusion 52 (2010) 000000 (16pp) UNCORRECTED PROOF

Dark-current-free petawatt laser-driven wakefieldaccelerator based on electron self-injection into anexpanding plasma bubble

S Y Kalmykov1,4, S A Yi1, A Beck2, A F Lifschitz3, X Davoine2,E Lefebvre2, V Khudik1, G Shvets1 and M C Downer1

1 The Department of Physics, The University of Texas at Austin, One University Station C1500,Austin, TX 78712, USA2 CEA, DAM, DIF, Bruyeres-le-Chatel, 91297 Arpajon Cedex, France3 Laboratoire de Physique des Gaz et des Plasmas, Universite Paris XI/CNRS, Batiment 210,91405 Orsay Cedex, France

E-mail: kalmykov@physics.utexas.edu

Received 11 November 2009, in final form 5 April 2010PublishedOnline at stacks.iop.org/PPCF/52

AQ1

AbstractA dark-current-free plasma accelerator driven by a short (�150 fs) self-guidedpetawatt laser pulse is proposed. The accelerator uses two-plasma layers, oneof which, short and dense, acts as a thin nonlinear lens. It is followed by along rarefied plasma (∼1017 electrons cm−3) in which background electrons aretrapped and accelerated by a nonlinear laser wakefield. The pulse overfocusedby the plasma lens diffracts in low-density plasma as in vacuum and drives inits wake a rapidly expanding electron density bubble. The expanding bubbleeffectively traps initially quiescent electrons. The trapped charge given byquasi-cylindrical three-dimensional particle-in-cell (PIC) simulations (usingthe CALDER-Circ code) is ∼1.3 nC. When laser diffraction saturates and self-guiding begins, the bubble transforms into a bucket of a weakly nonlinearnon-broken plasma wave. Self-injection thus never resumes, and the structureremains free of dark current. The CALDER-Circ modelling predicts a fewπ mm mrad normalized transverse emittance of electron beam accelerated inthe first wake bucket. Test-particle modelling of electron acceleration over9 cm (using the quasistatic PIC code WAKE) sets the upper limit of energy gain2.6 GeV with ∼2% relative spread.

(Some figures in this article are in colour only in the electronic version)

4 Author to whom any correspondence should be addressed. Present address: Department of Physics and Astronomy,University of Nebraska, Lincoln, NE 68588-0111, USA.

0741-3335/10/000000+16$30.00 © 2010 IOP Publishing Ltd Printed in the UK & the USA 1

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1. Introduction

Plasma wakefields driven by short (τL � 150 fs) petawatt (PW)-class laser pulses [1–6] canaccelerate electrons to gigaelectronvolt energy on top of an optical table [7, 8]. Presently, quasi-monoenergetic electrons in the energy range 150–300 MeV are routinely produced [9–17],and the gigaelectronvolt limit is approached with multi-terawatt lasers [12, 15, 18–20]. Newshort-pulse PW facilities (such as Texas Petawatt [3] (TPW) and POLARIS [4] lasers) openpossibilities beyond the gigaelectronvolt limit [21, 22]. High collimation and low-energyspread of gigaelectronvolts electrons needed for staged acceleration [23] and compact x-raysources [24–26] require optimization of electron injection in order to identify and eliminatephenomena causing ‘dark current’ (unwanted electrons trapped by the structure and acceleratedto high energy) [27].

The problem of dark current in modern laser wakefield accelerators (LWFAs) is aggravatedby the fact that to bypass the technical challenge of external injection they work in the blowout(or ‘bubble’) regime [28, 29]. Electrons accelerated in this regime are self-injected fromambient plasma into a single bucket of a strongly broken, fully electromagnetic plasma wake.The bucket forms when all plasma electrons facing the focused laser pulse are expelled byradiation pressure (while fully stripped ions remain immobile). The charge-separation fieldattracts bulk electrons to the axis thus forming a cavity devoid of electrons (bubble) that trailswith relativistic speed behind the driving laser. This structure has been recently visualized inoptical probing experiments [30]. In any transverse cross-section of the bubble, the acceleratinggradient is uniform and the focusing gradient is linear in radius [8, 31, 32], which is favourablefor emittance preservation. This structure, however, is not dark-current-proof. Uncontrollablevariations of shape and size of the laser-driven bubble occur in the course of propagation dueto both natural evolution of the driving pulse (focusing, defocusing, temporal compression)[21, 22, 33–37] and imperfections of the plasma target (such as long-scale inhomogeneities injet targets [38]). The resulting variations of wake potentials trigger self-injection of backgroundelectrons [21, 22, 33, 34–38]. Once the bubble size Lb grows by ∼1% over a propagation length∼10Lb, robust self-injection occurs even at very low densities, n0 ∼ 1017 cm−3 [21, 22, 36],where the Lorentz factor γb ∼ ω0/ωpe associated with ultrarelativistic bubble velocity is oforder 100 (here, ω0 is the laser frequency, ωpe = (4πe2n0/me)

1/2 is the electron plasmafrequency, −|e| and me are the electron charge and rest mass, respectively). Nonlinearbeam loading [39] further aggravates the situation leading to continuous uncontrollable self-injection [19, 20]. As a result, emittance dilution and degradation of electron spectra, such asgeneration of low-energy tails and multiple high-energy spikes, occur [13, 16, 19, 20, 37].

Conversely, a mildly nonlinear three-dimensional (3D) wake driven by a weakly focusedPW pulse in a strongly rarefied plasma (n0 ∼ 1017 cm−3, or ωpe/ω0 ∼ 0.01) [7, 40] isimmune to dark current. To take advantage of this regime, we propose to enforce self-injectionartificially by using a combination of bubble and weakly nonlinear wake in the same accelerator.The aim is to create the bubble locally, then let it rapidly expand until it transforms into the firstbucket of a conventional periodic 3D wake. The expanding bubble effectively traps electrons,which are further accelerated by the non-broken wake bucket with high collimation and afew per cent energy spread. This kind of wake evolution can be organized in a target thatconsists of two adjacent plasma layers of different densities. Importantly, in both plasmas theincident PW pulse is overcritical for relativistic self-focusing (RSF) [41]. A short dense slabplaced at the entrance plays the role of a thin nonlinear plasma lens (NLPL) [42]. RSF andthe focusing effect of electron density perturbations [43] impart a converging phase front yetkeep the waist of the pulse almost intact. Once the NLPL is optimized by appropriate choiceof thickness and electron density, the pulse released from the slab focuses in the low-density

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plasma without breaking up transversely. An electron density bubble forms in the vicinity ofthe nonlinear focus, then expands and traps copious electrons as the driving laser pulse diffractsout. Later on, a balance between linear diffraction and focusing nonlinearities is achieved,and the laser pulse with large spot size (rsg ∼ λp = 2π/kp, where kp = ωpe/c) and intensityIsg ∼ 1019 W cm−2 self-guides over many centimetres of the rarefied plasma. There it drivesa non-broken, dark-current-free plasma wake that accelerates earlier injected electrons.

The proposed layered plasma structure can be organized in a differentially pumped gas cell;plasma is created via optical field ionization. The cell may have two (or more) compartments ofadjustable length (from a few millimetres to many centimetres) held at different pressures. Theplasma thus consists of several adjacent uniform sections with millimetre length transitions(similar to the segmented capillary approach [44]). The large size of compartments andflexibility in NLPL parameters make the design robust.

The paper is organized as follows. In section 2 we overview the basic experimentalconfiguration of planned TPW experiments [21] in which electron self-injection inlongitudinally homogeneous plasmas and reduction of the dark current are difficult to achievetogether. This configuration and the layered target design presented in sections 3 and 4 mayserve as a reference for future experiments with similar laser systems [4]. In section 3, weintroduce the NLPL concept and model the dynamics of nonlinear overfocusing, diffraction andself-guiding of the PW laser pulse using a 3D cylindrically symmetric, fully relativistic, time-averaged particle-in-cell (PIC) code WAKE [45]. The simulations use variables (r, z, ξ), wherer = (x2 + y2)1/2 is the radius, z is the propagation variable and ξ/c = z/c − t is the retardedtime. The extended paraxial approximation used in the code enables precise calculation ofthe pulse group velocity and radiation absorption due to the creation of a plasma wake. Itapplies when the pulse waist size and length are many laser wavelengths. Self-injection andacceleration of plasma electrons are the subject of section 4. Section 4.1 outlines qualitativephysics of electron self-injection in the bubble driven by the diffracting laser pulse (moredetailed description can be found elsewhere [35, 36]). WAKE modelling of self-injectionand acceleration of test electrons in the quasistatic wake is discussed in section 4.2. Non-quasistatic test particles do not contribute to the electron density and current of the plasmawake in the WAKE modelling. The effect of beam loading which is critically important forthe determination of electron energy spread and, in some cases, responsible for saturationof self-injection [39], is thus ignored. To include this effect into the modelling frameworkin order to make experimentally meaningful predictions we resort to full 3D PIC modelling.Section 4.3 presents full 3D PIC simulations of electron self-injection dynamics via a newquasi-cylindrical code CALDER-Circ [46]. Electron self-injection during bubble expansion,termination of injection and formation of a monoenergetic bunch during bubble stabilizationand shrinkage are reproduced in a fully dynamic mode (with the beam loading effect included).The PIC simulations of section 4 prove that the optimized target design yields dark-current-free acceleration of 1.3 nC charge to 2.6 GeV energy with ∼2% relative energy spread and lessthan 8π mm mrad normalized transverse emittance. The conclusion summarizes the resultsand points out directions for further development. NLPL optimization is presented in theappendix.

2. Self-guiding of PW laser pulse in uniform plasmas

We present here one possible experimental configuration of a PW laser-driven plasmaaccelerator [4, 21]. A linearly polarized, Gaussian in space and time pulse with centralwavelength λ0 = 1.057 µm, average power P = 1.33 PW and full width at half maximumin intensity τL = 150 fs is focused to a spot with radius r0 = 80 µm (at the level exp(−2) of

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peak intensity) at the foot of a plasma density front ramp. It then propagates in the positive z

direction. The vacuum Rayleigh length ZR0 = (π/λ0)r20 ≈ 1.9 cm. The initial peak intensity

I0 = 1.35 × 1019 W cm−2 corresponds to normalized (to mec2/|e|) vector potential a0 ≈ 3.3.

A fully ionized helium plasma spans from z = 0 to 9.5 cm and has a flat-top density profilewith linear 1 mm long ramps at the entrance and exit. We model the laser self-guiding overthis distance using the code WAKE. The chosen grid size �ξ = (2/3)�r = 0.05k−1

p with 24macroparticles per radial cell is sufficient to resolve the fine structure of electron bubble.

Given the laser energy, duration and waist size (in the experiment they are prescribed bythe amplifier and focusing system), electron density remains the only parameter that controlsthe nonlinear pulse dynamics. The pulse with P/Pcr � 1 (where Pcr = 16.2ω2

0/ω2pe GW

is the RSF critical power [41]) and ωpeτL � π self-focuses until the charge-separation forceof a fully evacuated electron density channel (bubble with a radius Lb) balances the radialponderomotive force. Once the balance is achieved, the self-guided laser spot size is relatedto the normalized peak vector potential as kprsg = αa

1/2sg , where asg � 1. The value of

parameter α ≈ 2rsg/Lb depends on the regime of laser–plasma interaction. Simulations showthat 1.1 < α < 2 over a broad range of laser and plasma parameters [8, 47]. Using theexpression P/Pcr = a2

sg(kprsg)2/32 based on the fundamental Gaussian representation for the

radial profile of the laser pulse [41], we find asg = α−2/325/3(P/Pcr)1/3.

The effect of plasma density on pulse dynamics is examined for n0 = 5, 2.5 and1 × 1017 cm−3, which correspond to P/Pcr ≈ 40, 20 and 8, respectively. Figure 1(a)shows catastrophic consequences of nonlinear focusing in the highest density case, n0 =5 × 1017 cm−3 (ωpeτL ≈ 6). The pulse self-focuses by a factor >20 in intensity within7 mm (≈ZR0/3), breaks up transversely during defocusing and experiences rapid intensityoscillations during self-guiding. In addition, as shown in figure 1(b), nonlinear phase self-modulation and group velocity dispersion, as well as nearly 75% depletion due to the wakeexcitation, compress the pulse [8, 48] almost to a single cycle around z = 6 cm (we stop thesimulation soon after this point). The bubble radius in figure 1(e) is nearly twice the laser spotsize, which gives α ≈ 1, asg ≈ 11 and Isg ≈ 1.5×1020 W cm−2, in agreement with figure 1(a).The bubble is unable to close, which means that the quasistatic approximation fails for nearlyall macroparticles making up the bubble sheath. In these circumstances, a non-quasistatic 3DPIC simulation would show continuous electron self-injection.

For n0 = 2.5×1017 cm−3 (ωpeτL ≈ 4.2), self-focusing is milder, and figure 1(a) (red/darkgrey curve) shows that intensity increases only by a factor 8.5 after the first 1.2 cm (≈0.63ZR0).Transverse breakup is not observed, and the pulse self-guides with slowly varying peak intensityand moderate depletion (≈40% over 9.5 cm). The electron density snapshot in figure 1(f )shows that Lb/rsg ≈ 4/3, which gives α ≈ 3/2, asg ≈ 6.6 and Isg ≈ 5.4 × 1019 W cm−2, alsoin agreement with figure 1(a). This time, most plasma electrons remain quasistatic, and theelectron bubble allows for precise WAKE modelling. However, even modest pulsations of thepeak intensity and spot size of the self-guided pulse cause periodic variations of bubble size.Resulting periodic self-injection [21, 22] similar to that observed earlier in plasma channelsimulations [34] and in gas-jet experiments [37] leads to degradation of electron spectra.

For n0 = 1017 cm−3, peak intensity varies very steadily despite P/Pcr ≈ 8. The pulse frontsteepening seen in figure 1(d) is insignificant, and energy depletion remains below 5%. Fromfigure 1(g), rsg ≈ Lb, which gives α ≈ 2 (as in [8]), asg = kprsg = 4, Isg ≈ 2 × 1019 W cm−2.Not only does the incident pulse appear to be matched for self-guiding (r0 ≈ 1.175rsg), it isalso much shorter than a plasma period, ωpeτL = 2.7. RSF is thus additionally compensatedby nonlinear refraction caused by electron density perturbations [7, 49]. Although the pulseduration is almost resonant for longitudinal wake excitation [50], blowout does not fully

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Figure 1. Nonlinear focusing and self-guiding of the 1.33 PW laser pulse. Panel (a): peak intensityversus propagation distance for n0 = 1 (green/light grey), 2.5 (red/dark grey) and 5 × 1017 cm−3

(black). Panels (b), (c), (d): laser intensity in accordingly labelled positions in panel (a); dashedline is an iso-contour of laser pulse intensity at z = 0 (at exp(−2) of the peak). Panels (e), (f ),(g): first bucket of a plasma wake and laser intensity iso-contours corresponding to panels (b),(c), (d), respectively. Bottom: (h) peak intensity versus distance in very low-density plasma,n0 = 5 × 1016 cm−3; (i) electron density and intensity iso-contours for appropriately labelledpositions in panel (a). As laser diffracts out, wake becomes non-broken, and the first bucketgradually shrinks. (Colour online.)

develop (the density depression in the first bucket is about 80%). Steady evolution of thefirst bucket and the absence of complete blowout eliminate the threat of dark current; on theother hand, self-injection becomes problematic and has to be enforced artificially. Furtherreduction in electron density makes self-guiding inefficient, and the laser pulse diffracts out.

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When n0 = 5 × 1016 cm−3, a strong wakefield exists only over the distance L ≈ ZR0 [7, 49],as is clearly seen in figures 1(h) and (i).

Although dense plasmas (P/Pcr > 20, ωpeτL ≈ 2π ) perform poorly as guiding media,they can effectively focus high-intensity pulses. Focused intensity can be controlled withan appropriate choice of plasma density and length. On the other hand, rarefied plasmas(P/Pcr < 10, ωpeτL < 3) can support the dark current-free plasma wake over tens ofcentimetres with steady evolution and negligible laser depletion [7]. In the next section weshall combine the two plasmas in a single target that underlies a robust PW laser-driven darkcurrent-free LWFA with self-injection.

3. PW laser focusing with a thin NLPL

The target is composed of a short dense (nnlpl = 5 × 1017 cm−3) plasma slab (NLPL) followedby a long uniform plasma of much lower density, nacc = 1017 cm−3 (accelerator). The entireplasma length (NLPL and accelerator) is 9.5 cm, as in section 2. The density profile used insimulations is shown in inset (a) of figure 2. Technically, the dense slab may be an embeddedgas jet or a front compartment of a differentially pumped cell held at higher pressure. The pulsethat traverses the dense plasma acquires a converging phase front imparted by relativistic andponderomotive focusing nonlinearities [43]. As a result, the pulse focuses into the acceleratorplasma producing blowout, then diffracts and drives an expanding bubble that traps copiouselectrons. As soon as diffraction stabilizes and self-guiding begins, the bubble transformsinto a first non-broken bucket of a conventional nonlinear 3D plasma wake, and the electronself-injection ceases. Electrons trapped in the bucket are further accelerated with low energyspread and good collimation. To reduce aberrations, prevent transverse breakup of the pulseand avoid electron self-injection in the first bucket on the density downramp between the NLPLand accelerator plasmas [51], strong focusing and blowout inside the NLPL should be avoided(i.e. a thin lens approximation must hold [42]). This requirement limits the NLPL lengthto [43]

Lnlpl < (ZR0/2)(a0/2)2(P/Pcr)−1/2. (1)

For the 1.33 PW pulse of section 2, this condition gives Lnlpl < 4 mm, in agreement withWAKE modelling in the appendix. In the simulation presented below, the NLPL has a 4 mmflat-top portion and 1 mm linear ramps on either side (figure 2(a)). Figure 2 shows that thepulse focuses to peak intensity 1.15 × 1020 W cm−2 (spot size 24 µm) in 2.5 mm after NLPL.As shown in the appendix, peak focused intensity and the position of the nonlinear focus arealmost unaffected by the presence of low-density accelerator plasma. The NLPL, however, isnot free of aberrations, and about 40% of the pulse energy diffracts out of the box.

Comparison of the results of figure 2 with the cases of long uniform dense and rarefiedplasmas described in section 2 shows that the design with two-plasma layers uses benefits ofboth. The peak focused intensity can be as high as for the long dense uniform plasma, andcan be properly adjusted (by varying the NLPL thickness) in order not to disrupt subsequentself-guiding over ∼9 cm of low-density (accelerator) plasma. At the same time, the violentlaser dynamics typical of dense plasmas is completely avoided. This is the best case scenariofor the LWFA because the blowout and electron injection occur only once (in the vicinity ofnonlinear focus). The wake then remains mildly nonlinear and non-broken over the entireremaining propagation distance. We shall discuss details of electron injection and accelerationin the following section.

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Figure 2. Focusing and self-guiding of 1.33 PW laser pulse in a two-plasma target (density profileshown in inset (a)). Black and green/light grey dashed curves correspond to homogeneous plasmaswith n0 = 5 and 1 × 1017 cm−3, respectively. Red/dark grey curve corresponds to the pulseoverfocused with an NLPL (nnlpl = 5 × 1017 cm−3) and released for self-guiding into the rarefiedaccelerator plasma with nacc = nnlpl/5. Inset (b): peak intensity variation of the self-guidedlaser. Overfocusing does not destroy the pulse, which diffracts and then self-guides with just 40%depletion. (Colour online.)

4. Self-injection of electrons into the expanding plasma bubble and monoenergeticacceleration in dark-current-free LWFA

4.1. Qualitative physics of electron self-injection near nonlinear laser focus

In this and the following subsections, we discuss electron self-injection and acceleration usinga test-particle model. The model describes motion of initially quiescent test electrons inthe electromagnetic fields of self-consistently evolving quasistatic bubble (obtained from aWAKE simulation). A fully 3D test-particle tracking module built into the WAKE code is fullydynamic, relativistic and non-averaged in time. At each time step, a fresh group of quiescent(γe = 1) test electrons is placed in the simulation box before the laser pulse. The particletracker accurately describes their interaction with both quasi-paraxial high-frequency laserfield (taking into account the linear laser polarization) and slowly varying electric and magneticfields of the plasma wake [52]. For a given set of laser and plasma parameters, observation ofself-injection via test-particle modelling sufficiently motivates subsequent time-consumingmassively parallel 3D PIC simulations. Dynamics of electron injection and accelerationduring the period of bubble expansion and shrinkage (predicted in test-particle modelling)is reproduced in section 4.3 in a fully dynamic mode, with beam loading [39, 53] taken intoaccount, in a non-quasistatic 3D PIC simulation with the quasi-cylindrical CALDER-Circcode [46].

Once the PW pulse is focused by NLPL to intensity ∼1020 W cm−2 and spot sizerfoc ≈ 1.5k−1

p (as in figure 3(a)), the radiation pressure expels all electrons facing the pulse.Fully stripped heavy ions, however, remain at rest. The resulting charge-separation fieldattracts bulk electrons back to the axis. Trajectories of the innermost electrons overshoot. Theclosed electron density cavity surrounded by a dense shell (‘sheath’) of relativistic electronstrails behind the driving laser over the positive ion background. Electrons forming the sheathinteract with the bubble the longest [36]. When approaching the point of trajectory crossing,they are already pre-accelerated, γe = (1 − v2

e /c2)−1/2 � 1 [54] and, hence, have much

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Figure 3. Test electron injection in the expanding bubble and acceleration during the laser self-guiding. Top row—quasistatic electron density in units 1017 cm−3 (greyscale), the number densityof test electrons in (r, ξ) space (dots), and iso-contour of laser intensity (dashed line). Bottomrow—corresponding snapshots of longitudinal phase space of test electrons. Snapshots (a), (b), (c)and (d) are taken in the positions labelled a, b, c and d in figures 4(a) and (b). Panels (a) and (e):bubble and test electrons soon after the nonlinear focus; self-injection begins here. Panels (b) and(f ): the largest possible bubble and the longest electron bunch. Self-injection stops at this point andnever resumes. Panels (c) and (g) correspond to the point where the laser self-guiding begins. Firstbucket is no longer evacuated. Most of the test electrons are released from the bucket; the remainingparticles are further accelerated in the form of compact quasi-monoenergetic bunch. Snapshots (d)and (h) are taken half-way through the accelerator plasma where the laser is self-guided and firstwake bucket remains unchanged. (Colour online.)

higher inertia than the quasistatic electrons of the plasma bulk. As the laser diffracts afterthe nonlinear focus, the bubble expands and non-quasistatic test electrons get injected, as isclearly seen from a comparison of figures 3(a) and (b). The injection scenario is akin totrapping a relativistic projectile into the 3D potential bucket as it expands with time. If thebubble expands rapidly enough, some of the heavy sheath electrons lag behind the movingbubble boundary and stay inside the bubble. To trigger this effect, as established in full 3D PICsimulations for a broad range of parameters, the bubble should grow in size by ∼10% over afew tens of bubble lengths [21, 36]. A large fraction of injected electrons become trapped: theirmoving-frame (MF) Hamiltonian changes from HMF = γemec

2 − |e| − cPz = mec2 before

the arrival of the pulse to HMF < 0 (which would be impossible were the laser and bubblenon-evolving) [36]; here, is a slowly varying scalar potential and Pz is the longitudinalcomponent of the electron canonical momentum. The quasistatic approximation implies MFHamiltonian conservation [45]; thus, WAKE macroparticles cannot be trapped.

4.2. Self-injection and acceleration of test electrons in LWFA with NLPL

Figure 3 shows the laser pulse, first wake bucket and test electron phase space after the nonlinearfocus and during the guiding stage. Evolution of the laser peak intensity, the length of the firstbucket Lb (defined as the distance between the first potential maximum and the first minimumon axis) and initial positions and energy spectrum of trapped and accelerated test electrons aredisplayed in figure 4.

Snapshots (a) and (e) in figure 3 are taken immediately after the nonlinear focus (position ain figures 4(a) and (b)). The first wake bucket is a fully evacuated bubble. The bubble expansion

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Figure 4. Evolution of (a) laser peak intensity and (b) first wake bucket length (normalized to λp) inthe LWFA with a plasma lens. Positions labelled a, b, c and d correspond to panels (a), (b), (c) and(d) of figure 3. Inset in panel (a): longitudinal profile of electron density. Panel (c): longitudinalmomentum of test electrons at the exit plane (z = 9.5 cm) versus their initial positions. Inset:electron energy spectrum (red/dark grey and black correspond to the electrons from the first andsecond wake buckets, respectively). Comparison of panels (a), (b) and (c) shows that test electronsare self-injected during the laser defocusing and bubble expansion only. Subsequent shrinkageof the bubble stops the injection; injection never resumes during the self-guided stage. (Colouronline.)

has just begun, and signatures of self-injection are not clear yet. As the laser diffracts, thebubble grows and electron self-injection proceeds without interruption. It terminates onlywhen the bubble expansion stops at a distance ≈4.8 mm from the nonlinear focus (positionb in figures 4(a) and (b)). Figure 4(b) shows that the bubble length increases by 13.5% overthis distance (∼70 bubble lengths). The striking similarity between the number density of testelectrons and the quasistatic electron density in figures 3(a) and (b) shows that the majority ofelectrons remains quasistatic. Thus, wake evolution can be precisely modelled in a quasistaticframework (a similar situation was tested in [36]).

The bubble is largest and the test electron bunch longest, in figures 3(b) and (f ). After thispoint, the bubble contracts and stabilizes when the laser pulse becomes self-guided (positionc in figures 4(a) and (b)). Figures 3(c) and (g) correspond to this point. The first bucket is nolonger evacuated, and the wake restores its periodic structure (not shown). Contraction of thefirst bucket releases a large group of earlier injected electrons; most of them become trapped

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(with HMF < 0) in the second bucket (not shown). All electrons accelerated in the first twobuckets have HMF < mec

2.Snapshots (d) and (h) in figure 3 show that the wake bucket changes insignificantly

in the course of centimetre-long propagation, which facilitates test electron acceleration inthe form of a compact monoenergetic bunch. Finally, as is seen in panel (c) of figure 4,the bunch is accelerated to 2.6 GeV energy with a 2.35% energy spread. Normalizedtransverse emittance of the bunch (i.e. approximate area in the phase space r⊥p⊥) is εN,⊥ =(mec)

−1(〈r2⊥〉〈p2

⊥〉 − 〈r⊥p⊥〉2)1/2 ≈ 1.5π mm mrad. The electron bunch is concentrated nearthe axis where the focusing gradient is linear in radius. Therefore, as soon as the phase spaceof the bunch is filled and the bucket changes slowly, εN,⊥ is preserved [27]. Thus, the visiblyviolent injection process does not result in significant emittance dilution.

In this run, test electrons are seeded in the simulation box until the first bucket becomesnon-broken (z ≈ 2.1 cm). Initial positions of test particles reaching the end of the plasma(z = 9.5 cm) are plotted in figure 4(c). All these particles were trapped during the brief periodof laser diffraction and bubble expansion after the nonlinear focus (0.76 < z < 1.24 cm), andself-injection never resumed after that. The black dots in figure 4(c) show electrons releasedfrom the first bucket and trapped and accelerated in the second. Comparison with full 3D PICmodelling described in the next subsection shows that spectacularly strong contraction of thefirst bucket accompanied by massive release of earlier injected particles is overestimated inquasistatic modelling. Cylindrical beam loading suppresses this effect in full PIC simulation.But even in the worst case scenario copious electrons from the second bucket have averageenergy twice lower than the energy of the leading bunch, and two orders of magnitude largeremittance (due to ∼10 times higher angular divergence and spot size). Filtering them outexperimentally would be a minor technical problem.

4.3. Dynamics of electron self-injection in the quasi-cylindrical 3D PIC simulations

We complement the test-particle study of electron injection and acceleration with a fullydynamic simulation (beam loading included) using the recently developed quasi-cylindrical3D PIC code CALDER-Circ [46]. This code is highly efficient for the treatment of quasi-paraxial laser propagation in rarefied plasmas because it (a) preserves the realistic geometryof interaction and (b) accounts for the axial asymmetry by decomposing the electromagneticfields (laser and wake) into a few azimuthal modes (whereas the particles remain in full 3D).Thus, the 3D problem is reduced to an essentially 2D one. Reference [46] shows that in thecase of a linearly polarized laser, modes of order m � 2 contribute weakly to the electric field.Our restriction to modes m = 0 and m = 1 is therefore a very good approximation that allowsus to complete the simulation of figure 5 (1.5 cm of propagation) within 32 500 CPU hours(130 h on 250 processors). Resolution in the propagation direction is �z = 0.125c/ω0 and�r = 0.4λ0 is the radial grid size (total number of grid points 12 000 × 800 = 9.6 × 106); 15particles per cell provide enough accuracy to see both injection and beam loading. The timestep is �t = 0.124ω−1

0 . Reduced simulation load due to the favourable simulation geometry(well preserved axial symmetry) and reduced description of the radiation beam make it possibleto accomplish the quasi-cylindrical PIC modelling within time scales inaccessible to full 3DPIC codes.

The aim of our CALDER-Circ modelling is two-fold. First, as a part of codebenchmarking, we validate the dynamical behaviour of laser and bubble observed earlier in thequasi-paraxial, quasistatic WAKE modelling (e.g. features demonstrated in figure 3). Secondly,electron injection and acceleration during the period of bubble expansion and shrinkage(predicted in test-particle modelling with quasistatically evolving bubble) are reproduced

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Figure 5. Electron injection into expanding bubble in the quasi-cylindrical CALDER-Circsimulation. Top row: electron density (in units 1017 cm−3) in the plane orthogonal to the laserpolarization. Bottom row: corresponding longitudinal momentum distributions. Panels (a), (d)correspond to the propagation distance z = 0.83 mm; electrons with γ > ω0/ωpe show up. Panels(b), (e) correspond to z = 0.47 cm; here, the bubble is the largest, and injection stops. Panels(c), (f ) correspond to z = 1.5 cm, where the phase space rotation creates a monoenergetic bunch(energy spectrum in plot (g)). Panels (a), (b), (c) are direct counterparts of panels (a), (b), (c) offigure 3.

in fully dynamic mode, and with beam loading accounted for; this allows for meaningfulpredictions of the experimental outcome.

To explore the dynamics of electron injection into the expanding bubble, we start thesimulation with a linearly polarized Gaussian pulse focused at the plasma edge. The spot sizer0 = 24 µm and peak intensity 1.15 × 1020 W cm−2 correspond exactly to the pulse waist sizeand intensity in the nonlinear focus from the WAKE simulation of figures 3 and 4. Plasmadensity profile is flat-top with a 0.3 mm linear front ramp; n0 = 1017 cm−3 over the plateauregion. This numerical setup is designed to explore the laser defocusing after the nonlinearfocus in simulations of figures 3 and 4. Figures 5(a)–(c) show plasma density, and panels (d),(e) and (f )—the longitudinal phase space of electron bunch. Panel (g) displays the electronenergy spectrum at the end of the run, z = 1.5 cm. In panels 5(a) and (d), the bubble is shortest,and the signature of self-injection in phase space is just barely seen. These two panels arecounterparts of figures 3(a) and (e). In panels 5(b) and (e) the bubble is largest, and the electronbeam is longest. These panels are counterparts of figures 3(b) and (f ). Panels 5(c) and (f )correspond to figures 3(c) and (g) and show the first bucket after the stabilization.

Physical distances between the snapshots 5(a), (b) and (c) from the CALDER-Circ runand between their WAKE counterparts are the same: �za–b ≈ 3.87 mm and �zb–c ≈ 10.3 mm.Although dynamics of self-injection is very similar in both simulations, there are obviousdifferences that can be attributed to the phenomenon of nonlinear beam loading [39, 53].Comparison of fully self-consistent modelling (figure 5(b)) with its quasistatic counterpart(figure 3(b)) reveals noticeable distortion of the bubble shape near the rear end. The distortionis produced by transverse fields of the trapped electron bunch [39]. As a consequence, theaccelerating gradient reduces, and its longitudinal variation along the bunch becomes less steep.CALDER-Circ macroparticles in the front tip of the bunch are accelerated to the same energy astest electrons in the WAKE code, whereas the rest of the bunch gains on average 20% less energythan test particles. To assess the effect of beam loading we approximate the electron bunch from

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figure 5(b) with a flat-top distribution with a Gaussian radial profile, nb(r) = nb0 exp(−r2/σ 2b ),

where nb0 = (Qb/|e|)(πσ 2b lb)

−1 ≈ 2.35×1018 cm−3 is the peak electron density, Qb ≈ 1.5 nCis the total charge, σe ≈ 6 µm is the root-mean-square spot size and lb ≈ 35 µm is thelength of the bunch. According to [39], the sheath electrons cross the axis, and the bubbleremains closed until R4

b/(8r2t �0) > 1, where �0 = ∫ ∞

0 r(nb/n0) dr = (σ 2e /2)(nb0/n0) is

the normalized charge per unit length, rt is the bubble radius in the transverse cross-sectiontaken at the front tip of the bunch and Rb is the bubble radius in the central cross-section.Figure 5(b) gives �0 ≈ 420 µm2, rt ≈ 55 µm and Rb ≈ 71 µm. Hence, R4

b/(8r2t �0) ≈ 2.5,

and the bubble is not fully loaded. Moreover, CALDER-Circ simulation clearly indicatesthat injection continues without interruption until the moment of bubble stabilization andceases only when the contraction begins. Comparison of figures 5(b) and (c) clearly showsthat the transverse fields of the bunch are unable to preclude the bucket contraction; thebubble dynamics and the process of electron self-injection are thus governed primarily bythe evolution of the driver rather than by collective fields of trapped electrons. Self-injectionnever resumes after the point of bubble stabilization, which also agrees with the results ofsection 4.2.

As is seen in comparison of figures 3(c) and 5(c), the beam loading prevents strongcontraction of the bubble and massive release of trapped electrons into the second bucket(unlike that earlier observed in the quasi-static simulation). However, figures 5(e) and (f )show that the rear third of the bunch is truncated. Simultaneously, inhomogeneity of theaccelerating gradient leads to the phase space rotation: electrons injected later and situatednear the base of the bucket are exposed to the higher accelerating force and equalize inenergy with the head of the bunch soon after bucket stabilization (this is especially clearfrom comparison of figures 5(e) and (f )). The resulting bunch is monoenergetic with thecentral energy 460 MeV and relative energy spread 1.5% (figure 5(g)). This kind of phasespace rotation is different from that discussed in [55] and does not require driving the bunchuntil dephasing. Extrapolation of the obtained results to 7 cm acceleration distance givesE ∼ 2.5 GeV, which agrees with the upper limit set by test-particle modelling. Normalizedtransverse emittances, εN,i = (mec)

−1(〈x2i 〉〈p2

i 〉 + 〈xipi〉2)1/2, are εN,x = 6.5π mm mrad andεN,y = 8.2π mm mrad. As seen in figure 5(c), the first wake bucket is no longer an evacuatedbubble, hence, self-injection will not resume, and the low relative energy spread and emittanceare likely to be preserved.

5. Conclusion

The recently explained mechanism of self-injection of plasma electrons into an expandingelectron density bubble [36] is used as the basis for a new dark-current-free design of aPW laser-driven plasma accelerator. The design benefits from focusing properties of denseplasmas and guiding properties of rarefied plasmas. The proposed target consists of twouniform plasmas of different densities with a relatively sharp transition between them. Thethin (few millimetres) slab of dense plasma plays the role of a thin nonlinear lens [42]. Therelativistic focusing nonlinearity imparts a converging wave front onto the incident PW laserpulse while only moderately perturbing its waist. When released into the low-density plasma,the pulse focuses to intensity >1020 W cm−2 producing full electron blowout near the nonlinearfocus before rapidly diffracting. At this stage, the expanding electron density bubble trapscopious electrons. In the optimized design, diffraction finally saturates, and the laser pulseself-guides over about 9 cm of rarefied plasma, nacc = 1017 cm−3. It drives a mildly nonlinearplasma wake which accelerates ∼1.3 nC charge to the 2.6 GeV energy with a ∼2% spreadand <8π mm mrad normalized transverse emittance. The wake remains non-broken over the

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0 0.5 1 1.5 2 2.50

5

10

15

20

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mal

ized

peak

inte

nsity

0 1 2 3 40

2

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0.5

1

z (cm)

(a)

z (cm)

n 0 (10

17cm

–3

)

(b)

(d) (c)

z (cm)

Figure 6. Normalized peak intensity of the laser pulse (a), (b) and background electron density (c),(d) versus propagation length. Panels (a) and (c): focusing properties of the dense plasma slab asa function on its thickness (black: Lnlpl = 6 mm, red/dark grey: 4 mm (optimal), green/light grey:2 mm). Inset: the pulse overfocused with 6 mm slab is poorly guided compared with the optimalcase. Panels (b) and (d): the pulse focusing is weakly affected by the presence of low-density plasma(red/dark grey—two plasmas with Lnlpl = 4 mm (optimal case), black dashed—slab followed byvacuum). Inset shows where self-guiding begins in the low-density plasma. (Colour online.)

entire guided stage. Therefore, injection occurs only once (immediately after the nonlinearfocus) and dark current is eliminated. Further optimization of the scheme (in combinationwith a plasma channel [56]) has the potential to meet stringent beam quality requirements ofstaged acceleration and compact x-ray sources.

Acknowledgments

This work is supported by the US Department of Energy grants DE-FG02-04ER41321,DE-FG02-96ER40954 and DE-FG02-07ER54945. AB and EL acknowledge thesupport of LASERLAB-EUROPE/LAPTECH through EC FP7 contract no 228334.The authors acknowledge GENCI (www.genci.fr) and the Centre de Calcul pour laRecherche et la Technologie (www-ccrt.cea.fr) at CEA/Bruyere-le-Chatel for providinghigh performance computing resources that have contributed to the reported researchresults.

Appendix. Thin NLPL optimization

A 150 fs PW laser pulse (parameters specified in section 2) has to be focused in a low-densityaccelerator plasma (in our case, nacc = 1017 cm−3) to the spot size rfoc ∼ 1.5k−1

p ≈ 25 µmand intensity Ifoc > 1020 W cm−2. To this end, we use a NLPL—a thin slab of high-densityplasma (nnlpl � nacc); nnlpl is such that P/Pcr is of the order of a few tens, and plasma periodis close to the pulse duration. Nonlinear refraction [43] imparts a converging phase front ontothe weakly focused PW pulse, whereas the large spot size r0 ≈ 80 µm remains almost intact.The NLPL has to be optimized to prevent transverse breakup of the pulse during focusing andachieve stable self-guiding after defocusing.

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Longitudinal density profile of the target composed of NLPL and accelerator plasmas arepresented in figure 6(c). The length Lnlpl of the flat-top portion of the NLPL is restrictedby condition (1) which excludes laser self-focusing and electron blowout inside the slab.Aberrations are thus reduced, and transverse breakup of the pulse after NLPL is avoided.Density of accelerator plasma is chosen so as to keep P/Pcr < 10 and pulse durationmuch shorter than a plasma period. Mutual compensation of relativistic and ponderomotivenonlinearities [7, 49] facilitates vacuum-like focusing (one example is shown in figure 6(b))and increases the pulse stability against the relativistic filamentation [57]. The pulse focusesat a distance Lfoc from the edge of the NLPL, and then defocuses over roughly the samedistance until diffraction and nonlinear focusing compensate each other and self-guidingbegins.

WAKE simulations show that nnlpl = 5 × 1017 cm−3 is optimal for focusing the 1.33 PWpulse with the parameters specified in section 2 (P/Pcr = 40 and ωpeτL ≈ 6); for theseparameters equation (1) gives Lnlpl � 4 mm. Importantly, both Raman sidescatter andfilamentation [42] remain negligible over this short distance. Once the nnlpl is chosen, Lnlpl

prescribes the nonlinear focal length Lfoc and the peak focused intensity Ifoc. Figures 6(a) and(c) show that 2 mm increments of Lnlpl give a factor of 2 increase in Ifoc. Lnlpl can be easilycontrolled with this precision in a differentially pumped cell of adjustable length.

To achieve robust self-injection, not only has full electron blowout to be achieved in thevicinity of nonlinear focus, but the pulse should also rapidly diffract in order to cause ∼10%growth of the bubble size Lb over the distance Lfoc < 100Lb. The estimate Lb ≈ 0.65λp ≈70 µm from figure 4(b) sets the upper limit Lfoc < 7 mm. When Lnlpl < 1.5 mm, this limit isviolated. The bubble expands too steadily, and the number of injected electrons drops sharply.

Robust self-injection is achieved with Lnlpl in the range 2–4 mm. With Lnlpl = 2 mm,the NLPL aberrations are weak, and the laser is self-guided with higher intensity than in thecase Ls = 4 mm (as seen in the inset in figure 6(a)). Hence, strongly broken first wake bucketis maintained over almost 7 cm of propagation, and the acceleration proceeds entirely in thebubble regime. Average energy of accelerated test electrons is about 3.7 GeV with the relativespread about 15% and εN,⊥ ≈ 6.5π mm mrad.

Increasing Lnlpl to 4 mm reduces the focal length Lfoc from 6 to 2.5 mm and the spot sizefrom rfoc ≈ 32.5 to 24 µm. Focused peak intensity increases from 0.6 to 1.15 × 1020 W cm−2.Tighter focusing causes stronger aberrations which effectively reduce the laser power (40% oflaser energy diffracts out of the interaction region); the transverse beam breakup, however, isnot yet the case. Stages of self-injection in the bubble and acceleration in the weakly nonlinearwake become clearly distinguishable. In fact, the first wake bucket remains non-broken afterz ≈ 2 cm, which makes the structure essentially dark-current-free. This best case scenariounderpins the subject matter of this paper. Self-injected and accelerated test electrons have theenergy 2.6 GeV with 2.35% energy spread and εN,⊥ ≈ 1.5π mm mrad.

Further increase in Lnlpl results in the laser focusing inside the slab and violation of thethin lens approximation (black curves in figures 6(a) and (c)). For Lnlpl = 6 mm laser focusesto the spot rfoc = 17 mm ≈ k−1

p . Filamentation caused by the overfocusing [58] depletes thepulse energy in the interaction region by nearly 80%. Inset in figure 6(a) shows that the pulseis not effectively guided. Wakefield is weak, and the mean test electron energy is only 1.1 GeVwith the relative spread above 50%.

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